11)11] ON THE MOTION OF SOLID BODIES THROUGH VISCOUS LIQUID 39
Hydrodynamical solutions involving surfaces of discontinuity of the kind investigated by Hehnholtz and Kirchhoff provide indeed for a wake, but hero again there are difficult!UN. Behind a blade immersed transversely in a stream a region of "dead water" is indicated. The conditions of steady motion are thus satisfied; but, as Helmholtz himself pointed out, the motion thus defined is unstable. Practically the dead and live water are continually mixing; and if there be viscosity, the layer of transition rapidly assumes a finite width independently of the instability. One important; consequence is the development of a .suction on the, hind surface of the lamina which contributes in no insignificant degree to the total resistance. The amount of the suction does not appear to depend much on the degree of viscosity. When the latter is small, {.he. dragging action of the live npon the dead water extends to a greater distance, behind.
§ H. If the blade,, supposed infinitely thin, be moved edgeways through the fluid, the case, becomes one of "skin-friction." Towards determining the law of resistance Mr Lanchester hus put forward an argument* which, even if not rigorous, at any rate, throws an interesting light upon the question. Applied to the. case, of two dimensions in order to find the resistance F per unit length of blade, it, is somewhat as follows. Considering two systems for whieh the. velocity V of the blade, is different, let n be the proportional width of corresponding strata of velocity. The momentum communicated to the wake per unit length of travel is as nV, and therefore on the whole as nVy per unit of time. Thus /'' varies as nV*. Again, having regard to the law of viscosity and considering the strata contiguous to the blade, we sec; that F varies as Vfn. Hence, nV* varies aw Vfn, or V varies as w~a, from which it follows that F varies as V^l'1. If this be, admitted, the general law of dynamical similarity requires that for the whole* resistance
F**Qpv*l$ V1...............................(48)
when- / is the length, b the width of the blade, and G a constant. Mr Lanchentor gives thin in the form
F/p~ev*A*V*, .............................(49)
where A in the area of the lamina, agreeing with (48) if / and b maintain a constant ratio.
The difficulty in the way of accepting the above argument as rigorous is that complete similarity cannot be secured so long as b is constant as has boon supposed. If, as ia necessary to this end, we take /; proportional to n, it is bV/n, or V (and not V/n), which varies as nVa, or &7a. The conclusion is then simply that bV must be constant (v being given). This is merely the UBiial condition of dynamical similarity, and no conclusion as to the law of velocity follows.
* Aerodynamics, London, 1907, § 86.